mathematical problem

I almost agree.
In my first post I have 0.(9) * 10 - 0.(9) = 9.(9) - 0.(9), so it's not the second that's missing a nine, it's the first.
And the error in your post above would not be 0.09, but -0.09.
But basically is right. ;D

---------------------------------------------------------------------------------------------------------

And in our case the error is nailed flat to -0.000...0009, which is equal to 0 .

proof:

9.(9) - 0.(9) = 9 + 0.(9) - 0.(9) = 9 . (the first number has one nine less, after the point, than the second)

Let's scale it to an integral number. (with the most unusual factor)

9 + (∞/∞) * ( 0.(9) - 0.(9) ) = 9 ( both infinitys are the same )

=> 9 + (1/∞) * ( 99999...99990 - 99999...99999 ) = 9. (that both numbers got the same length)
=> 9 + (1/∞) * ( -9 ) = 9
=> 9 - (9/∞) = 9 - 9 * (1/∞) = 9

( the limit value of (1/∞) is 0 )

=> 9 - 9 * 0 = 9 - 0 = 9
=> 9 = 9 ( and the error is gone )